The Halo Occupation Distribution: Toward an Empirical Determination of the Relation between Galaxies and Mass

We investigate galaxy bias in the framework of the "halo occupation distribution" (HOD), which defines the bias of a population of galaxies by the conditional probability P(N|M) that a dark matter halo of virial mass M contains N galaxies, together with prescriptions that specify the relative spatial and velocity distributions of galaxies and dark matter within halos. By populating the halos of a cosmological N-body simulation using a variety of HOD models, we examine the sensitivity of different galaxy clustering statistics to properties of the HOD. The galaxy correlation function responds to different aspects of P(N|M) on different scales. Obtaining the observed power-law form of ξ_g(r) requires rather specific combinations of HOD parameters, implying a strong constraint on the physics of galaxy formation; the success of numerical and semianalytic models in reproducing this form is entirely nontrivial. Other clustering statistics such as the galaxy-mass correlation function, the bispectrum, the void probability function, the pairwise velocity dispersion, and the group multiplicity function are sensitive to different combinations of HOD parameters and thus provide complementary information about galaxy bias. We outline a strategy for determining the HOD empirically from redshift survey data. This method starts from an assumed cosmological model, but we argue that cosmological and HOD parameters will have nondegenerate effects on galaxy clustering, so that a substantially incorrect cosmological model will not reproduce the observations for any choice of HOD. Empirical determinations of the HOD as a function of galaxy type from the Two-Degree Field (2dF) and Sloan Digital Sky Survey (SDSS) redshift surveys will provide a detailed target for theories of galaxy formation, insight into the origin of galaxy properties, and sharper tests of cosmological models.